On eigenvalues of double branched covers

TL;DR

This paper investigates the minimal positive eigenvalues of double branched covers over knots, establishing bounds related to knot genus, dealternating number, and quasi-alternating properties, thereby linking spectral data to knot topology.

Contribution

It introduces a new spectral invariant based on eigenvalues of double branched covers and connects it to various knot invariants and properties, including non-orientable genus and quasi-alternating status.

Findings

The minimal number of positive eigenvalues bounds knot genus and related invariants.

Batson's bound for non-orientable 4-genus estimates the eigenvalue count.

A necessary condition for knots to be quasi-alternating is derived.

Abstract

For a given knot, we study the minimal number of positive eigenvalues of the double branched cover over spanning surfaces for the knot. The value gives a lower bound for various genera, the dealternating number and the alternation number of knots, and we prove that Batson's bound for the non-orientable 4-genus gives an estimate of the value. In addition, we use the value to give a necessary condition for being quasi-alternating.